GMAT – Quant

Manhattan Prep

Number Properties

Divisibility & Primes

Integers are ‘whole” numbers, such as 0, 1, 2, and 3, that have no fractional part. Integers can be positive(1, 2, 3...), negative (-1, -2, -3.. .), or the number 0.

Rules of Divisibility by Certain Integers2 if the integer is EVEN.

3 if the SUM of the integer’s DIGITS is divisible by 3.

4 if the integer is divisible by 2 TWICE, or if the LAST TWO digits are divisible by 4.

5 if the integer ends in 0 or 5.

6 if the integer is divisible by BOTH 2 and 3.

8 if the integer is divisible by 2 THREE TIMES, or if the LAST THREE digits are divisible by 8.

9 if the SUM of the integer’s DIGITS is divisible by 9.

10 if the integer ends in 0.

Factors and MultiplesFactors and Multiples are essentially opposite terms.A factor is a positive integer that divides evenly into an integer. 1, 2, 4 and 8 are all the factors (alsocalled divisors) of 8.A multiple of an integer is formed by multiplying that integer by any integer, so 8, 16, 24, and 32 aresome of the multiples of 8. Additionally, negative multiples are possible (-8, -16, -24, -32, etc.), butthe GMAT does not test negative multiples directly. Also, zero (0) is technically a multiple of everynumber, because that number times zero (an integer) equals zero.

Fewer Factors/ More Multiples

Divisibility and Addition/SubtractionIf you add or subtract multiples of N, the result is a multiple of N. You can restate this principle using any of the disguises above: for instance, if N is a. divisor of x and of y, then N is a divisor of x + y.

PrimesA prime number is any positive integer largerthan 1 with exactly two factors: 1 and itself. In other words, a prime number has no factors other than1 and itselfNote that the number 1 is not considered prime, as it has only one factor (itself). Thus, the first primenumber is 2, which is also the only even prime

Prime Factorizationif the problem states or assumes that a number is an integer, you may need to use prime factorizationto solve the problem.

Factor Foundation Ruleif a is a factor of by and b is a factor of c, then a is a factor of c. In other words, any integer is divisible by all of its factors—and it is also divisible by all of the factors of its factors.

The Prime BoxA Prime Box is exactly what its name implies: a box that holds all the prime factors of a number

Most of the time, when building a prime box for a variable, you will use a partial prime box, but whenbuilding a prime box for a number, you will use a complete prime box.Greatest Common Factor and Least Common MultipleGreatest Common Factor (GCF): the largest divisor of two or more integers.Least Common Multiple (LCM)s the smallest multiple of two or more integers.

Finding GCF and LCM Using Venn Diagrams

Be careful: even though you have no primes in the common area, the GCF is not 0 but 1.

RemaindersDividend = Quotient x Divisor + Remainderor Dividend = Multiple of Divisor + Remainder

Three Ways to Express RemaindersInteger Form / Fraction Form / Decimal Form17 = 3 x 5 + 2.

Ves

Creating Numbers with a Certain Remainder

Ch 2 – Odds, Evens, Positives & NegativesEven numbers are integers that are divisible by 2. Odd numbers are integers that are not divisible by 2.All integers are either even or odd.Evens: 0, 2, 4, 6, 8, 10, 12... Odds: 1, 3, 5, 7, 9, 11...Consecutive integers alternate between even and odd: 9, 10, 11, 12, 13...O, E, O, E, O...Negative integers are also either even or odd:Evens: -2, -4 , -6 , -8 , -10, -12... Odds: -1, -3, -5, -7, -9, -11...

Arithmetic Rules of Odds & Evens

The Sum of Two PrimesNotice that all prime numbers are odd, except the number 2. (All larger even numbers are divisible by2, so they cannot be prime.) Thus, the sum of any two primes will be even (“Add two odds . . .”), unlessone of those primes is the number 2. So, if you see a sum of two primes that is odd, one of those primesmust be the number 2. Conversely, if you know that 2 cannot be one of the primes in the sum, then thesum of the two primes must be even.

Representing Evens and Odd AlgebraicallyEven numbers are multiples of 2, so an arbitrary even number can be written as 2n, where n is any integer.Odd numbers are one more or less than multiples of 2, so an arbitrary odd number can be writtenas 2n + 1 or 2n — 1, where n is an integer.

Positives & NegativesNumbers can be either positive or negative (except the number 0, which is neither). A number line illustratesthis idea:Negative numbers are all to the left of zero. Positive numbers are all to the right of zero.

Absolute Value; Absolutely PositiveThe absolute value of a number answers this question: How far away is the number from 0 on thenumber line?|5|=5 & |-5|=5

Multiplying & Dividing Signed NumbersWe can summarize this pattern as follows: When you multiply or divide a group of nonzero numbers,the result will be positive if you have an even number of negative numbers. The result will be negative ifyou have an odd number of negative numbers.

Disguised Positives & Negatives Questions______

Some Positives & Negatives questions are disguised as inequalities. This generally occurs whenever aquestion tells you that a quantity is greater than or less than 0, or asks you whether a quantity is greaterthan or less than 0.

Ch-3 Combinatorics

The Words "OR" and "AND"OR = Add; AND = Multiply;Any time a question involves making decisions, there are twocases:Decision 1 OR Decision 2 (add possibilities)Decision 1 AND Decision 2 (multiply possibilities)

Arranging GroupsThe number of ways of arranging n distinct objects, if there are no restrictions, is nl (n factorial).

Arranging Groups with Repetition:The Anagram GridIf m members of a group are identical, divide the total number of arrangements by mlAnagramGrids can be used whenever you are arranging members of a group.The number of columns in the grid will always be equal to the number of members of the group.Rows = No. of Possibilities, No. of Options Available

Multiple GroupsSo far, our discussion of combinatorics has revolved around two main themes: (1) making decisions and(2) arranging groups. Difficult combinatorics questions will actually combine the two topics. In otherwords, you may have to make multiple decisions, each of which will involve arranging different groups.